Physics 263: Problem Set #18

1. (BTM 9.1.5 and 9.1.6.) For each of these example, the idea is to see if the third vector can be written as a₁ times the first plus a₂ times the second. In each case, there are three equations involving a₁ and a₂ that must be satisfied. See if they can be!
2. (BTM 9.2.1.) The exercise follows the example in eqs. (9.2.32) through (9.2.36), except that now the basis vectors are complex. That means you need to take the complex conjugate of them when you take the dot product with the vector |V>. The answers are in the back to check against.
3. (BTM 9.2.3.) Simply follow the example given in the text, first finding |1> by making |I> into a unit vector, then find |2'> as described, with |2> the corresponding unit vector, then finally |3'> as described, with unit vector |3>. There should be very little calculation.
4. (BTM 9.5.2 and 9.5.3.) First you need to identify the matrix R_z(theta). It is just the old rotation matrix in the upper left 2x2 block with a 1 in the diagonal lower right. Find the eigenvalues from det(R_z - wI) = 0 (the answers are in the back to check). Find the eigenvalues and eigenvectors for a variety of theta's using MATLAB and you'll find that the eigenvectors are independent of theta. For the second problem, just enter the matrices into MATLAB and find the eigenvalues and normalized eigenvectors using [V D] = eig(M), as described in last week's MATLAB cheatsheet.
5. Just follow the instructions. Note that you can search simultaneously for an author, a year, and (part of) a title.
6. (BTM 9.5.4 and 9.5.5.) For the first one, use the expansion of U as a power series in H (i.e., the definition of the matrix exponential) and in each term apply HV = hV. For the second one, write the elements explicitly as a = aR + i*aI, b = bR + i*bI, and so on, and derive conditions for the eight real numbers, reducing them to four independent ones in each case.